Lesson Least Squares Regression Line as Line of Best Fit

Transcription

1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION Comparing Lines for Predicting Textbook Costs In the previous lesson, you predicted the value of the response variable knowing the value of the explanatory variable (also known as predictor variable) using a best-fit line. In the homework you also investigated the concept of extrapolation, which is the idea that, even with the best line, the predictions based on this line may be unreliable if the value of the explanatory variable is outside the range of the data. So, how do you identify the line that is the best fit? You will use technology to find the equation of the line, but what does it mean to say that a particular line is the best fit? In this lesson, you will investigate this question with the goal of developing a method for determining which line is the best-fit line. 1 Here are the publishers suggested list prices in 2010 for 12 popular introductory statistics textbooks. The table below gives the descriptive statistics for the price data. Min Q1 Median Q3 Max Mean Standard Deviation List price A If someone asks you how much an introductory statistics textbook costs, what prediction would you give? Explain your reasoning. B What variables might be useful for predicting the cost of an introductory statistics textbook?

2 STATWAY STUDENT HANDOUT 2 C The number of pages in the textbook is one variable you could use to predict price. The scatterplot shows the relationship between pages and price for these 12 textbooks. The data have a somewhat linear form and the correlation coefficient is 0.79, so it makes sense to use a line to summarize the relationship between pages and price. Draw a line that you think is a good summary of the relationship between these two variables. Use the graph of your line to predict the price of a 650-page textbook. Then compare your prediction with a classmate. 2 Since there are infinitely many lines that you could draw, you need a way to determine which line is the best summary of the relationship between two quantitative variables. You will begin your investigation of how to define a best-fit line by comparing how well four lines predict the list price of the textbooks based on the number of pages.

3 Price Price Price Price STATWAY STUDENT HANDOUT 3 Line A (Mean Price) Line B Line C Line D A B Begin by using the equation for each line to complete the two incomplete rows in the table of predicted values. (You are predicting prices. It makes sense to write prices with two decimal places, such as $ instead of $ like you see in the table. You might be wondering why you are recording answers to four decimal places. This is because you will need this level of accuracy to develop some ideas later. So, record your answers to four decimal places for these activities.) Which of the four lines do you think results in the best overall predictions of price? Why? How are you selecting the best line?

4 STATWAY STUDENT HANDOUT 4 NEXT STEPS Thinking About Prediction Error 3 For each linear model, complete the missing parts of the table and answer the questions. A Line A (Mean Price) Identify the following textbooks in the scatterplot and the table: i The textbook for which the line comes closest to predicting the list price ii The textbook for which the prediction is furthest from the list price iii In the scatterplot, circle the textbooks that have a negative prediction error. What does a negative error tell you?

5 STATWAY STUDENT HANDOUT 5 B Line B Identify the following textbooks in the scatterplot and the table: i The textbook for which the line comes closest to predicting the list price ii The textbook for which the prediction is furthest from the list price iii How can you tell by looking at the scatterplot if the prediction error for a textbook is positive or negative? iv Identify a textbook for which Line A predicts too low a price but Line B predicts too high a price.

6 STATWAY STUDENT HANDOUT 6 C Line C Identify the following textbooks in the scatterplot and the table: i The textbook for which the line comes closest to predicting the list price ii The textbook for which the prediction is furthest from the list price iii All the textbooks for which the predicted list price is within $15 of the actual list price iv How can you tell by looking at the scatterplot that the prediction error is positive?

7 STATWAY STUDENT HANDOUT 7 D Line D Identify the following textbooks in the scatterplot and the table: i The textbook for which the line comes closest to predicting the list price ii the textbook for which the prediction is furthest from the list price iii all the textbooks for which the predicted list price is exceeds the actual list price by $20 or more E The goal is to identify a line that is the best summary of the relationship between pages and price. The best-fit line gives the best predictions of list price, which means that overall it has the least amount of error in the predictions. Rank the four lines from best to worst with the best being the line that gives the best overall predictions of list price. Briefly explain the reasoning behind your rankings.

8 STATWAY STUDENT HANDOUT 8 Which measures of the total error help you determine how well a line fits the data? Line Sum of Errors Sum of Absolute Value of Errors (SAE) Sum of Squares of Errors (SSE) A , B , C , D , Statisticians square the errors and then find the line that minimizes the sum of the squared errors. The line that has the smallest sum of the squared errors is called the least squares regression line. This line minimizes

9 Price STATWAY STUDENT HANDOUT 9 the sum of the squares of the errors, when compared to all other possible lines. You will use technology to find the equation of the least squares regression line. In the next lesson, you will learn more about the distinguishing features of this line. For these data, Line C is very close to the least squares regression line. Graphically they are indistinguishable. Here is the least squares line shown in the scatterplot. Now add the least squares line to your table of measurements for total error. Notice the following when you compare the least squares line to the other lines: The sum of the errors is zero for the least squares regression line. The sum of the squares of the errors is the smallest for this line (hence the name least squares). This is true if you compare the least squares regression line to any other line you created. The sum of the absolute value of the errors is not the smallest for the least squares regression line. Line C is a better fit if you use this criterion. Which measures of the total error help determine how well a line fits the data? Line Sum of Errors SAE SSE A , B , C , D , Least squares regression line , Now let s return to the criteria you developed to identify best-fit lines at the beginning of the lesson. Given the discussion today, which of the criteria you developed are not true for the least squares line? Which seem to be valid criteria given your work today?

10 STATWAY STUDENT HANDOUT 10 TAKE IT HOME 1 Here you have data collected from students at Los Medanos College in The variable units gives the number of college course units the student reported he or she was taking that semester. The variable textbooks gives the amount that the student reported spending on textbooks or other resources required for their courses that semester. A Use technology to find the least squares regression line. (Think carefully about which variable is the explanatory variable.) B Use the least squares regression line to predict the amount spent on textbooks for a student taking 12 units. C Explain why the least squares regression line is considered the line of best fit. 2 With the following applet, you can draw a line that you think fits the data well and compare your line to the least squares regression line. Note: In the applet, errors are called residuals. This term comes from thinking about a data point as composed of two parts: the part explained by the regression line (the prediction) and the part that is leftover (called the residual or error).

11 STATWAY STUDENT HANDOUT 11 A Instructions 1) Check Your line and click Move line. Follow directions to move the line so that it fits the data well. 2) Check Show residuals and record the SAE for your line in the table below. 3) Check Show squared residuals and record the SSE for your line in the table. 4) Check Regression line. 5) Check Show residuals and record the SAE for the regression line in the table. 6) Check Show squared residuals and record the SSE for the regression line in the table. Line Predicting Height Based on Foot Length Equation of Line SAE SSE Your line Regression line B Compare the values of the SAE and SSE for your line with the regression line. What do you notice? Why does this make sense? C When you click Show residuals, you see vertical line segments drawn from each data point to the regression line. Some line segments are long and others are short. Why is this? Why do you think these vertical line segments are shown?

12 STATWAY STUDENT HANDOUT 12 D When you click Show squared residuals, you see squares appear. Some squares are small and others are large. Why is this? Why do you think these squares are shown? This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0 of Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.

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